The origin and implications of dark matter anisotropic cosmic infall on ~L* haloes

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The origin and implications of dark matter anisotropic

cosmic infall on L* haloes

Dominique Aubert, Christophe Pichon, Stephane Colombi

To cite this version:

Dominique Aubert, Christophe Pichon, Stephane Colombi. The origin and implications of dark matter

anisotropic cosmic infall on L* haloes. Monthly Notices of the Royal Astronomical Society, Oxford

University Press (OUP): Policy P - Oxford Open Option A, 2004, 352, pp.376-398.

�10.1111/j.1365-2966.2004.07883.x�. �hal-00008398�

The origin and implications of dark matter anisotropic cosmic infall

on

≈ L
haloes

D. Aubert,

 C. Pichon

and S. Colombi

3_{Numerical Investigations in Cosmology (NIC), CNRS, France}

Accepted 2004 March 26. Received 2004 March 26; in original form 2003 December 22

A B S T R A C T

We measure the anisotropy of dark matter flows on small scales (∼500 kpc) in the near environment of haloes using a large set of simulations. We rely on two different approaches to quantify the anisotropy of the cosmic infall: we measure the flows at the virial radius of the haloes while describing the infalling matter via fluxes through a spherical shell; and we measure the spatial and kinematical distributions of satellites and substructures around haloes detected by the subclump finderADAPTAHOPdescribed for the first time in the appendix. The two methods are found to be in agreement both qualitatively and quantitatively via one- and two-point statistics.

The peripheral and advected momenta are correlated with the spin of the embedded halo at levels of 30 and 50 per cent. The infall takes place preferentially in the plane perpendicular to the direction defined by the spin of the halo. We computed the excess of equatorial accretion both through rings and via a harmonic expansion of the infall.

The level of anisotropy of infalling matter is found to be∼15 per cent. The substructures have their spin orthogonal to their velocity vector in the rest frame of the halo at a level of about 5 per cent, suggestive of an image of a flow along filamentary structures, which provides an explanation for the measured anisotropy. Using a ‘synthetic’ stacked halo, it is shown that the positions and orientations of satellites relative to the direction of spin of the halo are not random even in projection. The average ellipticity of stacked haloes is 10 per cent, while the alignment excess in projection reaches 2 per cent. All measured correlations are fitted by a simple three-parameter model.

We conclude that a halo does not see its environment as an isotropic perturbation, we investigate how the anisotropy is propagated inwards using perturbation theory, and we discuss briefly the implications for weak lensing, warps and the thickness of galactic discs.

Key words: galaxies: formation – galaxies: haloes – dark matter.

1 I N T R O D U C T I O N

Isotropy is one of the fundamental assumptions in modern cos-mology and is widely verified on very large scales, both in large galaxy surveys and in numerical simulations. However, on scales of a few megaparsecs, the matter distribution is structured in clusters and filaments. The issue of anisotropy down to galactic and clus-ter scales has long been studied, as it is related to the search for large-scale structure in the near environment of galaxies. For exam-ple, both observational studies (e.g. West 1994; Plionis & Basilakos 2002; Kitzbichler & Saurer 2003) and numerical investigations (e.g.

E-mail: aubert@astro.u-strasbg.fr

Faltenbacher et al. 2002) showed that galaxies tend to be aligned with their neighbours and support the vision of anisotropic merg-ers along filamentary structures. On smaller scales, simulations of rich clusters showed that the shape and velocity ellipsoids of haloes tend to be aligned with the distribution of infalling satellites, which is strongly anisotropic (Tormen 1997). However, the point is still moot and recent publications did not confirm such an anisotropy using resimulated haloes; they proposed 20 per cent as a maximum for the anisotropy level of the distribution of satellites (Vitvitska et al. 2002).

When considering preferential directions within the large-scale cosmic web, the picture that comes naturally to mind is one involv-ing these long filamentary structures linkinvolv-ing large clusters to one other. The flow of haloes within these filaments can be responsible

for the emergence of preferential directions and alignments. Previ-ous publications showed that the distributions of spin vectors are not random. For example, haloes in simulations tend to have their spin pointing orthogonally to the direction of the filaments (Faltenbacher et al. 2002). Furthermore, down to galactic scales, the angular mo-mentum remains mainly aligned within haloes (Bullock et al. 2001). Combined with the results suggesting that the spins of haloes are mostly sensitive to recent infall (van Haarlem & van de Weygaert 1993), these alignment properties fit well with accretion scenarios along special directions: angular momentum can be considered as a good marker to test this picture.

Most of these previous studies focused on the fact that alignments and preferential directions are consequences of the formation pro-cess of haloes. However, the effects of such preferential directions on the inner properties of galaxies have been less addressed. It is widely accepted that the properties of galaxies partly result from their interactions with their environments. While the amplitude of the interactions is an important parameter, some issues cannot be studied without taking into account the spatial extension of these interactions. For example, a warp may be generated by the torque imposed by infalling matter on the disc (Ostriker & Binney 1989; L´opez-Corredoira, Betancort-Rijo & Beckman 2002): not only the direction but also the amplitude of the warp are a direct consequence of the spatial configuration of the perturbation. Similarly, it is likely that disc thickening due to infall is not independent of the incom-ing direction of satellites (e.g. Quinn, Hernquist & Fullagar 1993; Huang & Carlberg 1997; Velazquez & White 1999).

Is it possible to observe the small-scale alignment? In particular, weak lensing deals with effects as small as the level of detected anisotropy (if not smaller) (e.g. Croft & Metzler 2000; Heavens, Refregier & Heymans 2000; Hatton & Ninin 2001); hence it is im-portant to put quantitative constraints on the existence of alignments on small scales. Therefore, the present paper also addresses the is-sue of detecting preferential projected orientations on the sky of substructures within haloes.

Our main aim is to provide quantitative measurements to study the consequences of the existence of preferential directions on the dynamical properties of haloes and galaxies, and on the observation of galaxy alignments. Hence our point of view is more galactocen-tric (or cluster-cengalactocen-tric) than previous studies. We search for local alignment properties on scales of a few hundred kiloparsecs. Using a large sample of low-resolution numerical simulations, we aim to ex-tract quantitative results from a large number of halo environments. We reach a higher level of statistical significance while reducing the cosmic variance. We applied two complementary approaches to study the anisotropy around haloes: the first one is particulate and uses a new substructure detection toolADAPTAHOP; the other one is the spherical galactocentric fluid approach. Using two methods, we can assess the self-consistency of our results.

After a brief description of our set of simulations (Section 2), we describe the galactocentric point of view and study the properties of angular momentum and infall anisotropy measured at the virial radius (Section 3). In Section 4 we focus on anisotropy in the dis-tribution of discrete satellites and substructures, and we study the properties of the satellites’ proper spins, which provide an explana-tion for the detected anisotropy. In Secexplana-tion 5 we discuss the level of anisotropy as seen in projection on the plane of the sky. We then investigate how the anisotropic infall is propagated inwards and dis-cuss the possible implications of our results to weak lensing and to the dynamics of the disc through warp generation and disc thicken-ing (Section 6). Conclusions and prospects follow. The Appendix describes the substructures detection toolADAPTAHOPtogether with

the relevant aspects of one-point centred statistics on the sphere. We also formally derive there the perturbative inward propagation of infalling fluxes into a collisionless self-gravitating sphere.

2 S I M U L AT I O N S

In order to achieve a sufficient sample and ensure convergence of the measurements, we produced a set of∼500 simulations. Each of them consists of a 50 h−1 Mpc3 _{box containing 128}3_particles.

The mass resolution is 5× 109_M

. A CDM cosmogony (m=

0.3, _ = 0.7, h = 0.7 and σ8 = 0.928) is implemented with

different initial conditions. These initial conditions were produced withGRAFIC(Bertschinger 2001), where we chose a BBKS (Bardeen et al. 1986) transfer function to compute the initial power spectrum. The initial conditions were used as inputs to the parallel version of the tree codeGADGET(Springel, Yoshida & White 2001b). We set the softening length to 19 h−1kpc. The halo detection was performed using the halo finderHOP(Eisenstein & Hut 1998). We employed the density thresholds suggested by the authors (outer= 80, δsaddle=

2.5δouter,δpeak= 3δouter) As a check, we computed the halo mass

function at z= 0 defined as (Jenkins et al. 2001):

f (σ(M)) = M ρ0

dn

d lnσ−1. (1)

Here n(M) is the abundance of haloes with a mass less than M and

ρ0is the average density, whileσ2(M) is the variance of the density

field smoothed with a top-hat filter at a scale that encloses a mass M. The simulations mass function is shown in Fig. 1 and compared to the Press–Schechter model (see Press & Schechter 1974) and to the fitting formula given by Jenkins et al. (2001). The Press–Schechter model overestimates the number of small haloes by a factor of 1.7 as already demonstrated by, for example, Gross et al. (1998). The fitting formula seems to be in better agreement with the measured mass function with an accuracy of∼10 per cent for masses below 3× 1014_M

.

As another means to check our simulations and to evaluate the convergence ensured by our large set of haloes, we computed the probability distribution of the spin parameterλ, defined as (Bullock et al. 2001) −1.0 −0.5 0.0 0.5 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 PS Jenkins et al. 2001 −1.0 −0.5 0.0 0.5 −3.0 −2.5 −2.0 −1.5 −1.0 ln(f) ln(σ1₎ ln(σ1₎ ∆ ln(f)/ln(f)

Figure 1. Top: the mass function f (σ(M)) of haloes (thin full line)

com-pared to the Press–Schechter model (thick dashed line) and to the fitting formula of Jenkins et al. (2001) (thick full line). Bottom: relative residuals between the fitting formula and the mass function.

λ’0=0.035 σ=0.57 λ’0=0.0347 σ=0.629 0.00 0.05 0.10 0.15 0 5 10 15 20 λ’ P( λ ’)

Figure 2. The distribution of the spin parameter λ defined asλ ≡ J/(√2M V R200) computed using 100 000 haloes with a mass greater than

5× 1012_M

. The distribution can be fitted with a log-normal function with parametersλ₀= 0.0347 ± 0.0006 and σ = 0.63 ± 0.02 (solid line). The curve parametrized byλ₀= 0.035 and σ = 0.57 is also shown (dashed line). The two results are almost coincident, indicating that the value ofσ is not so strongly constrained using a log-normal distribution.

λ_{≡ √} J

2M V R200

. (2)

Here J is the angular momentum contained in a sphere of virial radius

R200 with a mass M and V2 = GM/R200. The measurement was

performed on 100 000 haloes with a mass larger than 5× 1012_M

 as explained in the next section. The resulting distribution forλis shown in Fig. 2. The distribution P(λ) is well fitted by a log-normal distribution (e.g. Bullock et al. 2001):

P(λ) dλ= 1 λ√_2πσ exp
−ln2(λ/λ0) 2σ2  dλ. (3)

We foundλ₀= 0.0347 ± 0.0006 and σ = 0.63 ± 0.02 as best-fitting values, consistent with the parameters (λ= 0.035 and σ = 0.57) found by Peirani, Mohayaee & De Freitas Pacheco (2004), but our value ofσ is slightly larger. However, using σ = 0.57 does not lead to a significantly different result. The value of σ is not strongly constrained and no real disagreement exists between our and their best-fitting values. The halo’s spin, on which some of the following investigations are based, is computed accurately.

3 A G A L AC T O C E N T R I C P O I N T O F V I E W The analysis of exchange processes between the haloes and the in-tergalactic medium will be carried out using two methods. The first one can be described as ‘discrete’. The accreted objects are explic-itly counted as particles or particle groups. This approach will be applied and discussed later in this paper. The other method relies on measuring directly relevant quantities on a surface at the interface between the halo and the intergalactic medium. In this approach, the measured quantities are scalar, vector or tensor fluxes, and we assign to them flux densities. The flux density representation allows us to describe the angular distribution and temporal coherence of infalling objects or quantities related to this infall. The formal rela-tion between a flux density,(Ω), and its associated total flux, ,

through a region S is

≡

 S

(Ω)· dΩ, (4)

whereΩ denotes the position on the surface where  is evaluated and dΩ is the surface element normal to this surface. Examples of flux densities are mass flux density,ρvr, or accreted angular

mo-mentum,ρvr·L. In particular, this description in terms of a spherical

boundary condition is well suited to study the dynamical stability and response of galactic systems. In this section, these fields are used as probes of the environment of haloes.

3.1 Halo analysis

Once a halo is detected, we study its environment using a galac-tocentric point of view. The relevant fields(Ω) are measured on the surface of a sphere centred on the halo’s centre of mass with radius R200[where 3M/(4πR3200)≡ 200ρ] (cf. Fig. 3). There is no

exact, nor unique, definition of the halo’s outer boundary and our choice of R200(also called the virial radius) is the result of a

com-promise between a large distance to the halo’s centre and a good signal-to-noise ratio in the determination of spherical density fields. We used 40× 40 regularly sampled maps in spherical angles Ω = (ϑ, φ), allowing for an angular resolution of 9◦. We take into ac-count haloes with a minimum number of 1000 particles, which gives a good representation of high-density regions on the sphere. This minimum corresponds to 5× 1012_M

 for a halo, and allows us to reach a total number of 10 000 haloes at z= 2 and 50 000 haloes at

z= 0. This range of mass corresponds to a somewhat high value for

a typical L_galaxy but results from our compromise between reso-lution and sample size. Detailed analysis of the effects of resoreso-lution is postponed to Aubert & Pichon (2004).

The density,ρ(Ω), on the sphere is computed using the particles located in a shell with a radius of R200and a thickness of R200/10

(this is quite similar in spirit to the counts-in-cells techniques widely used in analysing large-scale structures, but in the context of a sphere the cells are shell segments). Weighting the density with quantities such as the radial velocity or the angular momentum of each particle contained within the shell, the associated spherical fields,ρvr(Ω)

orρL(Ω), can be calculated for each halo. Two examples of spher-ical maps are given in Fig. 3. They illustrate a frequently observed discrepancy between the two types of spherical fields,ρ(Ω) and

ρvr(Ω). The spherical density field,ρ(Ω), is strongly quadrupolar,

which is due to the intersection of the halo triaxial three-dimensional density field by our two-dimensional virtual sphere. By contrast the flux density of matter,ρvr(Ω), does not have such a quadrupolar

distribution. The contribution of halo particles to the net flux density is small compared to the contribution of particles coming from the outer intergalactic region.

3.2 Two-point statistics: advected momentum and the halo’s spin

The influence of infalling matter on the dynamical state of a galaxy depends on whether or not the infall occurs inside or outside the galactic plane. If the infalling matter is orbiting in the galactic plane, its angular momentum is aligned with the angular momentum of the disc. Taking the halo’s spin as a reference for the direction of the ‘galactic’ plane, we want to quantify the level of alignment of the orbital angular momentum of peripheral structures (i.e. as measured on the virial sphere) relative to that spin. The inner spin

S is calculated using the positions and velocities (rpart,vpart) of the

NGP SGP 0 l= NGP SGP l 0

Figure 3. A galactocentric point of view of the density field,ρ(Ω) (top),

and of the flux density of mass,ρvr(Ω), surrounding the same halo (bot-tom). This measurement was extracted from aCDM cosmological simu-lation. The considered halo contained about 1013M_{ or 2000 particles. The} high-density zones are darker. The density’s spherical field shows a strong quadrupolar component with high-density zones near the two poles while this component is less important for the mass flux density field measured on the sphere. This discrepancy between the two spherical fields is common and reflects the shape of the halo as discussed in the main text.

particles inside the R200 sphere in the centre-of-mass rest frame

(r0,v0):

S=

part

(rpart− r0)× (vpart− v0). (5)

Here r0is the position of the halo centre of mass, whilev0stands

for the average velocity of the halo’s particles. This choice of rest frame is not unique; another option would have been to take the most bounded particle as a reference. Nevertheless, given the considered mass range, no significant alteration of the results is to be expected. The total angular momentum, LT (measured at the virial radius,

R200) is computed for each halo using the spherical fieldρL(Ω):

LT=



4π

ρL(Ω) · dΩ. (6)

The angle,θ, between the spin of the inner particles S and the total orbital momentum LTof ‘peripheral’ particles is then easily

computed: θ = cos−1  LT·S |LT||S|  . (7)

Measuring this angleθ for all the haloes of our simulations al-low us to derive a raw probability distribution of angle, dr(θ). An

isotropic distribution corresponds to a non-uniform probability den-sity diso(θ). Typically diso is smaller near the poles (i.e. near the

region of alignment), leading to a larger correction for these angles and to larger error bars in these regions (see Fig. 4): this is the con-sequence of smaller solid angles in the polar regions (which scales like∼sin θ) than in equatorial regions for a given θ aperture. The true anisotropy is estimated by measuring the ratio:

dr(θ)/diso(θ) ≡ 1 + ξL S(θ), (8)

Here, 1+ ξLS(θ) measures the excess probability of finding S and

LTaway from each other, whileξLS(θ) is the cross-correlation of the

angles of S and LT. Thus havingξLS(θ) > 0 (respectively, ξLS(θ) < 0)

implies an excess (respectively, a lack) of configurations with aθ separation relative to an isotropic situation.

To take into account the error in the determination ofθ, each count (or Dirac distribution) is replaced with a Gaussian distribution and contributes to several bins:

δ(θ − θ0)→ N (θ0, σ0)= 1 σ0 √ 2πexp
−(θ − θ0)2 2σ2 0  , (9) whereN stands for a normalized Gaussian distribution and where the angle uncertainty is approximated byσ0∼ (4π/N)1/2using N

particles as suggested by Hatton & Ninin (2001). If Nvis equal to

the number of particles used to computeρL(Ω) on the virial sphere and if Nhis the number of particles used to compute the halo’s spin,

the error we associated to the angle between the angular momentum at the virial sphere and the halo’s spin is

σ0=



(4π/Nv)+ (4π/Nh)∼



(4π/Nv), (10)

because we have Nv  Nh. Note that this Gaussian correction

introduces a bias in mass: a large infall event (large Nv, smallσ0)

is weighted more for a givenθ0than a small infall (small Nv, large

σ0). All the distributions are added to give the final distribution:

dr(θ) =

Np

 p

N (θp, σp), (11)

where Npstands for the total number of measurements (i.e. the total

number of haloes in our set of simulations). The corresponding isotropic angle distribution is derived using the same set of errors randomly redistributed: diso(θ) = Np  p Nθiso p , σp . (12)

Fig. 4 shows the excess probability, 1+ ξLS(θ), of the angle

between the total orbital momentum of particles at the virial radius LTand the halo’s spin S. The solid line is the correlation deduced

from 40 000 haloes at redshift z= 0. The error bars were determined using 50 subsamples of 10 000 haloes extracted from the whole set of available data. An average Monte Carlo correlation and a Monte Carlo dispersionσ is extracted. In Fig. 4, the symbols stand for the average Monte Carlo correlation, while the vertical error bars stand for the 3σ dispersion.

The correlation in Fig. 4 shows that all angles are not equivalent sinceξLS(θ) = 0. It can be fitted with a Gaussian curve using the

following parametrization: 1+ ξL S(θ) = a1 √ 2πa3 exp  −θ − a22 2a2 3  + a4. (13)

The best-fitting parameters are a1= 2.351 ± 0.006, a2= −0.178 ±

0.002, a3 = 1.343 ± 0.002 and a4 = 0.6691 ± 0.0004. The 

ρL ρvrL ρvrL fit ρL fit 0.8 1.0 1.2 1.4 Θ (rad) 1+ ξ (Θ )

0

π/4

π/2

3

π/4

π

Figure 4. Excess probability, 1+ ξLS(θ), of the angle, θ, between the halo’s spin (S) and the angular momentum (LTfor total, or LAfor accreted)

measured on the virial sphere using the fluid located at the virial radius. Here LT represents the total angular momentum measured on the virial

sphere (solid line and circles) and LAthe total accreted angular momentum

measured on the sphere (dashed line and diamonds). The error bars represent the 3σ dispersion measured on subsamples of 10 000 haloes. The correlation takes into account the uncertainty on the angle determination due to the small number of particles at the virial radius. HereξLS(θ) ≡ 0 would be expected for an isotropic distribution of angles between S and L while the measured distributions indicate that the aligned configuration (θ ∼ 0) is significantly more likely. The two excess probability distributions are well fitted by Gaussian functions (almost coincident red curves in Synergy: see main text).

maximum being located at small angles, the aligned configura-tion, LT·S ∼ 0, is the most enhanced configuration (relative to

an isotropic distribution of angleθ). The aligned configuration of LT relative to S is 35 per cent [ξLS(0)= 0.35], more frequent in

our measurements than for a random orientation of LT. As a

conse-quence, matter is preferentially located in the plane perpendicular to the spin, which is hereafter referred to as the ‘equatorial’ plane.

The angles, (ϑ, φ), are measured relative to the z- and x-axes of the simulation boxes and not relative to the direction of the spin. Thus we do not expect artificial LT–S correlations due to the sampling

procedure. Nevertheless, it is expected on geometrical grounds that the aligned configuration is more likely since the contribution of

recent infalling dark matter to the halo’s spin is important. As a

check, the same correlation was computed using the total advected orbital momentum:

LA=



4π

Lρvr(Ω)· dΩ. (14)

The resulting correlation (see Fig. 4) is similar to the previous one but the slope towards small values of θ is even stronger and for example the excess of aligned configuration reaches the level of 50 per cent [ξLS(0)∼ 0.5]. The correlation can be fitted following

equation (13) with a1 = 3.370 ± 0.099, a2 = −0.884 ± 0.037,

a3 = 1.285 ± 0.016 and a4 = 0.728 ± 0.001. This enhancement

confirms the relevance of advected momentum for the build-up of the halo’s spin, though the increase in amplitude is limited to 0.2 for

θ = 0. The halo’s inner spin is dominated by the orbital momentum

of infalling clumps (given the larger lever arm of these virialized clumps and their high radial velocities) that have just passed through the virial sphere, as suggested by Vitvitska et al. (2002) (see also

Appendix D). It reflects a temporal coherence of the infall of matter and thus of angular momentum, and a geometrical effect: a fluid clump that is just being accreted can intersect the virtual virial sphere, being in part both ‘inside’ and ‘outside’ the sphere. Finally a small fraction of the accreted momentum may come from ma-terial that has already passed once through the R200 sphere. This

component would be aware of the dynamical properties of the inner halo. Thus it is expected that the halo’s spin S and the momenta LTand LAat the virial radius are correlated since the halo’s spin

is dominantly set by the properties of the angular momentum in its outer region. The anisotropy of the two fields LT and LA do not

have the same implication. The spatial distribution of advected an-gular momentum, LA, contains stronger dynamical information. In

particular, the variation of the angular momentum of the halo plus disc is induced by tidal torques but also by accreted momentum for an open system. For example, the anisotropy of Lρvr should be

reflected in the statistical properties of warped discs as discussed later in Sections 6.1 and 6.2.

3.3 One-point statistics: equatorial infall anisotropy

The previous measurement does not account for dark matter falling into the halo with a very small angular momentum (radial orbits). We therefore measured the excess of equatorial accretion,δm, defined

as follows. We can measure the average flow density of matter, r,

in a ring centred on the equatorial plane:

r ≡ 1 Sr  −π/8<θ−π/2<π/8 ρvr(Ω)· dΩ, (15) where Sr = 

−π/8<θ−π/2<π/8dΩ. The ring region is defined by

the area where the polar angle satisfiesθpol = π/2 ± π/8, which

corresponds to about 40 per cent of the total covered solid angle. The larger this region is, the better the convergence of the average value of r, but the lower the effects of anisotropy, since averaging

over a larger surface leads to a stronger smoothing of the field. This value of±π/8 is a compromise between these two contradictory trends. In the next section and in the Appendix, we discuss more general filtering involving spherical harmonics that are related to the dynamical evolution of the inner component of the halo. We also measure the flow averaged over all the directions :

≡ ρvr ≡ 1 4π  4π ρvr(Ω)· dΩ. (16)

Since we are interested in accretion, we computed r and using

only the infalling part of the density flux of matter, whereρvr(Ω)·

dΩ< 0, ignoring the outflows. The fraction of outflowing material decreases from 20 per cent of the total integrated flux at z= 0 to 10 per cent at z= 2. We define δmas

δm≡

r−

. (17)

This number quantifies the anisotropy of the infall. It is positive when infall is in excess in the galactic equatorial plane, while for isotropic infallδm≡ 0. The quantity δmcan be regarded as being

the ‘flux density’ contrast of the infall of matter in the ring region (formally it is the centred top-hat-filtered mass flux density contrast as shown in Appendix C1). This measurement, in contrast to those of the previous section, does not rely on some knowledge of the inner region of the halo but only on the properties of the environment.

Fig. 5 displays the normalized distribution ofδmmeasured for

50 000 haloes with a mass in excess of 5× 1012_M

 and for dif-ferent redshifts (z= 1.8, 1.5, 0.9, 0.3, 0.0). The possible values for

z=0. z=0. z=0.33 z=0.9 z=1.3 z=1.7 −1.0 −0.5 0.0 0.5 1.0 0.2 0.4 0.6 0.8 1.0 1.2 δm PDF( δm) z=0. z=0.33 z=0.9 z=1.3 z=1.7 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 −0.2 −0.1 0.0 0.1 0.2 δm 0.5(PDF( δm )– PDF( –δ m ))

Figure 5. Top: normalized probability distributions (PDF) of the excess of

equatorial infall,δm, measured at the virial radius. The quantity 1+ δmstands

for the ratio between the flux of matter through the equatorial subregion of the R200sphere and the average flux of matter through the whole R200sphere.

The equatorial subregion is defined as being perpendicular to the direction of the halo’s spin. It formally corresponds to the top-hat-smoothed mass flux density contrast. The valueδm= 0 is expected for an isotropic infall of

matter through the virial sphere. The average value ofδmis always greater

than zero, indicating that the infall of matter is, on average, more important in the direction orthogonal to the halo’s spin vector than in other directions. Bottom: the antisymmetric part of theδmdistribution. Being positive for

positive values ofδm, the antisymmetric part of theδmdistributions shows

that accretion in the equatorial plane is in excess relative to that expected from isotropic accretion of matter.

δmrange betweenδm∼ −1 and ∼1.5. The average value δm of

the distributions is statistically larger than zero (see also Fig. 6). Here stands for the statistical expectation, which in this paper is approximated by the arithmetic average over many haloes in our simulations. The antisymmetric part of the distribution ofδmis

pos-itive for pospos-itiveδm. The probability distribution function (PDF) of

δmis skewed, indicating an excess of accretion through the

equa-torial ring. The median value forδmisδmed= 0.11, while the first

25 per cent haloes haveδm< δ25≡ −0.11 and the first 75 per cent

haloes haveδm< δ75≡ 0.37. Therefore we have (δ75− δmed)/(δ25−

δmed)= 1.13, which quantifies how the distribution of δmis

posi-tively skewed. The skewness S3= (δ− ¯δ)3/(δ− ¯δ)23/2is equal to

0.44. Combined with the fact that the average valueδm is always

linear fit 3 σ error 0.0 0.5 1.0 1.5 0.14 0.15 0.16 0.17 0.18 redshift z <δ m >(z)

Figure 6. The redshift evolution ofδm. The average  is performed on

a set of 40 000 haloes at z= 0 and 10 500 haloes at z = 1.8. The error bars stand for the error on the estimation ofδm with  = σ(δm)/

√ N , where N is the number of haloes needed to computeδm. The value of δm is

always positive and indicates an excess of accretion in the equatorial plane. This redshift evolution can be fitted asδm (z) = 0.0161(± 0.0103)z +

0.147(± 0.005). This excess is detected for every redshift smaller than z = 2, which indicates an excess of accretion in the equatorial region. We applied a mass threshold of 5× 1012_M

 to our haloes for every redshift. Then, the halo population is different from one redshift to another. This selection effect may dominate the observed time evolution.

positive, this shows that the infall of matter is larger in the equatorial plane than in the other directions.

This result is robust with respect to time evolution (see Fig. 6). At redshift z= 1.8, we have δm = 0.17, which falls to δm = 0.145

at redshift z= 0. This redshift evolution can be fitted as δm (z) =

0.0161(± 0.0103)z + 0.147(± 0.005). This trend should be taken with caution. For every redshift z we take in account haloes with a mass bigger than 5× 1012_M

. Thus the population of haloes studied at z= 0 is not exactly the same as the one studied at z = 2. Actually, at z= 0, there is a strong contribution of small haloes (i.e. with a mass close to 5× 1012_M

) that have just crossed the mass threshold. The accretion on small haloes is more isotropic as shown in more details in Appendix D2. One possible explanation is that they experienced less interactions with their environment and have since had time to relax, which implies a smaller correlation with the spatial distribution of the infall. Also bigger haloes tend to lie in more coherent regions, corresponding to rare peaks, whereas smaller haloes are more evenly distributed. The measured time evolution of the anisotropy of the infall of matter therefore seems to result from a competition between the trend for haloes to become more symmetric and the bias corresponding to a fixed mass cut.

In short, the infall of matter measured at the virial radius in the direction orthogonal to the halo’s spin is larger than expected for an isotropic infall.

3.4 Harmonic expansion of anisotropic infall

As mentioned earlier (and demonstrated in Appendix A), the dynam-ics of the inner halo and disc is partly governed by the statistical properties of the flux densities at the boundary. Accounting for the gravitational perturbation and the infalling mass or momentum re-quires projecting the perturbation over a suitable basis such as the

spherical harmonics:

(Ω) =

,m

αm

Ym(Ω). (18)

Here, stands for the mass flux density, the advected momentum flux density, or the potential perturbation, for example. The result-ingαm

 coefficients correspond to the spherical harmonic

decompo-sition in an arbitrary reference frame. The different m correspond to the different fundamental orientations for a given multipole. A spherical field with no particular orientation gives rise to a field av-eraged over the different realizations that appear as a monopole, i.e. αm = 0 for  = 0. Having constructed our virial sphere in a

refer-ence frame attached to the simulation box, we effectively performed a randomization of the orientation of the sphere. However, since the direction of the halo’s spin is associated to a general preferred orien-tation for the infall, it should be traced through theα_mcoefficients. Let us define the rotation matrix,R, which brings the z-axis of the simulation box along the direction of the halo’s spin. The spherical harmonic decomposition centred on the spin of the halo, am

, is

given by (e.g. Varshalovich, Moskalev & Khersonskii 1988):

am  = R  αm  ≡ m Rm,m , (ϑ, ϕ)αm_. (19)

If the direction of the spin defines a preferential plane of accretion, the corresponding am

 will be systematically enhanced. We therefore

expect the equatorial direction (which corresponds to m = 0 for every) not to converge to zero.

We computed the spherical harmonic decomposition ofρvr(ϑ,

ϕ) given by equation (18) for the mass flux density of 25 000 haloes

at z= 0, up to  = 15. For each spherical field of the mass density flux, we performed the rotation that brings the halo’s spin along the z-direction to obtain a set of ‘centred’ am

 coefficients. We also

computed the related angular power spectra C_:

C_≡ 1 4π 1 2 + 1   m=− am  2 = 1 4π 1 2 + 1   m=− _αm  2 . (20) Let us define the normalized ˜am

 (or harmonic contrast, see

Appendix C1), ˜a_m≡√4πa m  a0 0 = am signa0 0 √ C0 . (21)

This compensates for the variations induced by our range of masses for the halo. For each , we present in Fig. 7 the median value, |Re{˜am

}| for  = 2, 4, 6, 8 computed for 25 000 haloes. All the

˜am

 have converged towards zero, except for the ˜a0 coefficients. The imaginary parts of ˜am

 have the same behaviour, except for the

Im{˜a_0} coefficients, which vanish by definition (not shown here). The m= 0 coefficients are statistically non-zero. We find ˜a0

2 =

−0.65 ± 0.04, ˜a0

4 = 0.12 ± 0.02, ˜a60 = −0.054 ± 0.015 and

˜a0

8 = 0.0145 ± 0.014. Errors stand for the distance between the

5th and the 95th percentile. The typical pattern corresponding to an m = 0 harmonic is a series of rings parallel to the equatorial plane. This confirms that accretion occurs preferentially in a plane perpendicular to the direction of the halo’s spin.

The spherical accretion contrast δ[ρvr](ϑ, φ) can be

recon-structed using the˜am

 coefficients (as shown in the Appendix):

δ[ρvr](ϑ, ϕ) =

 ,m

˜a_mY_m(ϑ, ϕ) − 1. (22) In Fig. 8, the polar profile

 δ[ρvr](ϑ)  ≡ ,m  ˜am   Ym (ϑ, 0) − 1 (23) −5 0 5 0.0 0.5 −0.05 0.00 0.05 0.10 0.15 0.00 0.02 0.04 0.06 −5 0 5 0.00 0.01 0.02

|<a

>|

|<a

>|

|<a

>|

m

|<a

>|

Figure 7. The convergence of the modulus of the real part of˜am , for  = 2, 4, 6, 8. The ˜am

 decomposition was computed for 25 000 haloes, and each coefficient has been normalized with the corresponding C0(see text

for details). Here, stands for the median while the error bars stand for the distance between the 5th and 95th percentiles. The median value of˜am

 is

zero except for the˜a0

 coefficient: this is a signature of a field invariant to azimuthal rotations. L<=5 L<=15 0.3 0.2 0.1 0.0 0.1 0.2

0

π/4

π/2

3

π/4

π

Figure 8. An illustration of the convergence of ˜a_mpresented in Fig. 7. The solid line stands for the azimuthal average of the spherical contrast of accretion computed using equation (23), the dotted line for the spherical field reconstructed with
5. The insert represents the reconstructed spherical field using the expansion of the ˜a_mof the mass flux measured at the virial sphere. The sphere presents an excess of accretion in the equatorial region because of the non-zero average value of ˜a_0coefficients (for even values of).

of this reconstructed spherical contrast is shown. This profile has been obtained using the˜am

 coefficients with 
5 and 
15.

The contrast is large and positive nearϑ = π/2 as expected for an equatorial accretion. The profile reconstructed using
5 is quite similar to the one using
15. This indicates that most of the energy involved in the equatorial accretion is contained in a typical angular scale of 36◦ (a scale that is significantly larger thanπ/20 corresponding to the cut-off frequency in our sampling of the sphere as mentioned earlier).

Using a spherical harmonic expansion of the incoming mass flux density (equation 8), we confirmed the excess of accretion in the equatorial plane found above. This similarity was expected since these two measurements (using a ring or using a spherical harmonic expansion) can be considered as two different filterings of the spher-ical accretion field as demonstrated in Appendix C. The main asset of the harmonic filtering resides in its relevance for the description of the inner dynamics as discussed in Section 6.

3.5 Summary

To sum up, the two measurements of Sections 3.2 and 3.3 (or 3.4) are not sensitive to the same effects. The first measurement (in-volving the angular momentumρL at the virial radius) is mostly a measure of the importance of infalling matter in building the halo’s proper spin. The second and the third measurements (involving the excess of accretion in the equatorial plane,δm, using rings and

har-monic expansion) are quantitative measures of coplanar accretion. The equatorial plane of a halo is favoured relative to the accretion of matter (compared to an isotropic accretion) to a level of∼12 per cent between z= 2 and z = 0. Down to the halo scale (∼500 kpc), anisotropy is detected and is reflected in the spatial configuration of infalling matter.

4 A N I S OT R O P I C I N FA L L O F S U B S T R U C T U R E S

To confirm and assess the detected anisotropy of the matter infall on haloes in our simulations, let us now move on to a discrete framework and measure related quantities for satellites and substructures. In the hierarchical scenario, haloes are built up by successive mergers of smaller haloes. Thus if an anisotropy in the distribution of infalling matter is to be detected, it seems reasonable that this anisotropy should also be detected in the distribution of satellites. The previous galactocentric approach for the mass flow does not discriminate be-tween an infall of virialized objects and a diffuse material accretion, and therefore is also sensitive to satellites merging: one would need to consider, say, the energy flux density. However, it is not clear if satellites are markers of the general infall and Vitvitska et al. (2002) did not detect any anisotropy at a level greater than 20 per cent.

The detection of substructures and satellites is performed us-ing the codeADAPTAHOP, which is described in detail in the Ap-pendix. This code outputs trees of substructures in our simulations, by analysing the properties of the local dark matter density in terms of peaks and saddle points. For each detected halo we can extract the whole hierarchy of subclumps or satellites and their characteristics. Here we consider the leaves of the trees, i.e. the most elementary substructures that the haloes contain. Each halo contains a ‘core’, which is the largest substructure in terms of particle number, and ‘satellites’, corresponding to the smaller ones. We call the ensemble of core plus satellites the ‘mother’ or the halo. Naturally the number of substructures is correlated with the mother’s mass. The bigger the number of substructures, the bigger the total mass. Because the

resolution in mass of our simulations is limited, smaller haloes tend to have only one or two satellites. Thus in the following sections we will discriminate cases where the core has less than four satellites. A total of 50 000 haloes have been examined, leading to a total of about 120 000 substructures.

4.1 Core spin–satellite orbital momentum correlations In the mother–core–satellite picture, it is natural to regard the core as the central galactic system, while satellites are expected to join the halo from the intergalactic medium. One way to test the effect of large-scale anisotropy is to compare directly the angle between the core’s spin, Sc, and the satellites’ angular momentum, Ls, relative to

the core. These two angular momenta are chosen since they should be less correlated with each other than, for example, the halo’s spin and the angular momentum of its substructures. Furthermore, par-ticles that belong to the cores are strictly distinct from those that belong to satellites, thus preventing any ‘self-contamination’ effect. As a final safeguard, we took into account only satellites with a dis-tance relative to the core larger than the mother’s radius. The latter quantity is computed using the mean square distance of the parti-cles belonging to the mother, and thus we focus only on ‘external’ satellites. The core’s spin is

Sc=

 p

(rp− rc)× (vp− vc), (24)

where rp andvp (respectively rc andvc) stand for the particles’

positions and velocities (respectively the core’s centre-of-mass po-sition and velocity) and where

rp < dc, (25)

where dcis the core’s radius. The angular momentum for a satellite is

computed likewise, with a different selection criterion on particles, namely

|rp− rs| < ds, (26)

where rsstands for the satellite’s centre-of-mass position and dsis

its radius.

Fig. 9 displays the reduced distribution of the angle,θcs, between

the core’s spin and the satellites’ orbital momentum, whereθcs is

defined by θcs= cos−1  Ls·Sc |Ls||Sc|  . (27)

The Gaussian correction was applied as described in Section 3.2, to take into account the uncertainty on the determination ofθcs.

The correlation ofθcsindicates a preference for the aligned

con-figuration with an excess of∼12 per cent of aligned configurations relative to the isotropic distribution. We ran Monte Carlo realiza-tions using 50 subsamples of 10 000 haloes extracted from our whole set of substructures to constrain the error bars. We found a 3σ er-ror of 6 per cent: the detected anisotropy exceeds our erer-rors, i.e.

ξcs(θcs) is not uniform with a good confidence level. The variations

with the fragmentation level (i.e. the number of satellites per sys-tem) remains within the error bars. The best-fitting parameters for the measured distributions of systems with at least one satellite are

a1 = 0.3993 ± 0.0038, a2 = 0.0599 ± 0.0083, a3 = 0.8814 ±

0.0055 and a4= 0.9389 ± 0.0002 (see equation 13 for

parametriza-tion). Not surprisingly, a less structured system shows a stronger alignment of its satellites’ orbital momentum relative to the core’s spin. In the extreme case of a binary system (one core plus a satel-lite), it is common for the two bodies to have similar masses. Since

Nsat>0 Nsat>3 Nsat>10 fit Nsat>0 0.95 1.00 1.05 1.10 1.15 Θ(rad) 1+ ξ (Θ )

0

π/4

π/2

3

π/4

π

3-

σ errors

Figure 9. Excess probability, 1+ ξcs(θcs), of the angle between the core’s

spin and the orbital momentum of satellites. Cores have at least one satel-lite (solid line), four satelsatel-lites (dashed line) and 10 satelsatel-lites (dotted line). These curves have been normalized by the expected isotropic distribution and the Gaussian correction was applied to account for errors on the angle determination. Hereξcs(θ) = 0 is expected for an isotropic distribution of

angles between the core’s spin and the orbital momentum. All satellites are external to the core, yet an excess of alignment is present. The triangles represent the angle distribution, the error bars stand for the 3σ dispersion for 50 subsamples of 10 000 satellites (out of 35 000) while the dash-dotted curve (red in the online version of this article on Synergy) stands for the best Gaussian fit of the distribution for systems with at least one satellite (see equation 13 for parametrization). The best-fitting parameters are: a1=

0.3993± 0.0038, a2= 0.0599 ± 0.0083, a3= 0.8814 ± 0.0055 and a4=

0.9389± 0.0002. The isotropic case is excluded with a good confidence level, even for systems with a large number of satellites.

the two bodies are revolving around each other, a natural preferen-tial plane appears. The core’s spin will be likely to be orthogonal to this plane. Increasing the number of satellites increases the isotropy of the satellites’ spatial distribution (the distribution’s maxima are lower and the slope towards low values ofθcsis gentler), but

switch-ing from at least four satellites to at least 10 satellites per system does not change significantly the overall shape distribution. This suggests that convergence, relative to the number of satellites, has been reached for theθcsdistribution.

As the measurements of the anisotropy factorδmindirectly

sug-gested, satellites have an anisotropic distribution of their directions around haloes. Furthermore the previous analysis of the statistical properties ofδ (Section 3.3) indicated an excess of aligned config-uration of 15 per cent, which is consistent with the current method using substructures. While the direction of the core’s spin should not be influenced by the infall of matter, we still find the existence of a preferential plane for this infall, namely the core’s equatorial plane.

4.2 Satellite velocity–satellite spin correlation

The previous sections compared the properties of haloes with those of satellites. In a galactocentric framework, the existence of this preferential plane could only be local. In the extreme each halo would then have its own preferential plane without any connection to the preferential plane of the next halo. Taking the satellite itself as a reference, we have analysed the correlation between the satellite’s

average velocity in the core’s rest frame and the structure’s spin. Since part of the properties of these two quantities are consequences of what happened outside the galactic system, the measurement of their alignment should provide information on the structuration on scales larger than the halo scale, while sticking to a galactocentric point of view.

For each satellite, we extract the angle,θvs, between the velocity

and the proper spin and derive its distribution using the Gaussian correction (see Fig. 10). The satellite’s spin Ssis defined by

Ss=

 p

(rp− rs)× (vp− vs), (28)

where rsandvsstands for the satellite’s position and velocity in the

halo core’s rest frame. The angleθvsbetween the satellite’s spin and

the satellite’s velocity is

θvs= cos−1  Ss·vs |Ss||vs|  . (29)

Only satellites external to the mother’s radius are considered while computing the distribution of angles. This leads to a sam-ple of about 40 000 satellites, at redshift z= 0. The distribution

ξ(θvs) was calculated as sketched in Section 2. An isotropic

distri-bution ofθvswould as usual lead to a uniform distributionξ(θvs)=

0. The result is shown in Fig. 10. The error bars were computed using the same Monte Carlo simulations described before with 50 subsamples of 10 000 satellites.

We obtain a peaked distribution with a maximum forθvs= π/2

corresponding to an excess of orthogonal configuration of 5 per cent compared to a random distribution of satellite spins relative to their

Nsat>0 Nsat<3 Nsat>3 fit Nsat>0 0.95 1.00 1.05 Θ (rad) 1+ ξvs (Θ )

0

π/4

π/2

3

π/4

π

3-

σ errors

Figure 10. Excess probability, 1+ ξvs, of the angle between the

sub-structures’ spin and their velocities in the mother’s rest frame. The Gaus-sian correction was applied to take into account uncertainty on the angle determination. The distribution was measured for all mothers (solid line), mothers with at least four substructures (dotted line) and mothers with at most three substructures (dashed line). The triangles represent the mean an-gle distribution. The error bars represent the Monte Carlo 3σ dispersion for 50 subsamples of 10 000 haloes (out of 35 000). The dash-dotted curve (red in the online version of this article on Synergy) stands for the best fit of the distribution with a Gaussian function for systems with at least one satellite (see equation 13 for parametrization). The best-fitting parameters are: a1=

0.2953± 0.0040, a1= 1.5447 ± 0.0015, a2= 0.8045 ± 0.0059 and a3=

0.9144± 0.0010. In the core’s rest frame, the satellites’ motion is orthogo-nal to the direction of the satellites’ spin. This configuration would fit in a picture where structures move along filamentary directions.

velocities. The substructure’s motion is preferentially perpendicular to their spin. This distribution of angles for systems with at least one satellite can be fitted by a Gaussian function with the following best-fitting parameters (see equation 13): a1 = 0.2953 ± 0.0040,

a1= 1.5447 ± 0.0015, a2= 0.8045 ± 0.0059 and a3= 0.9144 ±

0.0010. The variation with the mother’s fragmentation level is within the error bars. However, the effect of an accretion orthogonal to the direction of the spin is stronger for satellites that belong to less structured systems. This may again be related to the case where two comparable bodies revolve around each other, but from a satellite point of view. The satellite spin is likely to be orthogonal to the revolution plane and consequently to the velocity’s direction.

This result was already known for haloes in filaments (Faltenbacher et al. 2002), where their motion occurs along the fila-ments with their spins pointing outwards. The current results show that the same behaviour is measured down to the satellite’s scale. However, this result should be taken with caution since Monte Carlo tests suggest that the error (deduced from the 3σ dispersion) is about 4 per cent.

This configuration where the spins of haloes and satellites are orthogonal to their motion fits with the image of a flow of structures along the filaments. Larger structures are formed out of the merging of smaller ones in a hierarchical scenario. Such small substructures should have small relative velocities in order eventually to merge while spiralling towards each other. The filaments correspond to regions where most of the flow is laminar, hence the merging be-tween satellites is more likely to occur when one satellite catches up with another, while both satellites move along the filaments. During such an encounter, shell crossing induces vorticity perpendicular to the flow as was demonstrated in Pichon & Bernardeau (1999). This vorticity is then converted to momentum, with a spin orthogonal to the direction of the filament.

Finally, the flow of matter along the filaments may also provide an explanation for the excess of accretion through the equatorial regions of the virial sphere. If a sphere is embedded in a ‘laminar’ flow, the density flux detected near the poles should be smaller than that detected near the ‘equator’ of the sphere. The flux measured on the sphere is larger in regions where the normal to the surface is collinear with the ‘laminar’ flow, i.e. the ‘equator’. On the other hand, a nil flux is expected near the poles since the vector normal to the surface is orthogonal to the direction of the flow. The same effect is measured on Earth, which receives the Sun’s radiance: the temperature is larger in the tropics than near the poles. Our observed excess of accretion through the equatorial region supports the idea of a filamentary flow orthogonal to the direction of the halo’s spin down to scales
500 kpc.

5 P R O J E C T E D A N I S OT R O P Y

5.1 Projected satellite population

We looked directly into the spatial distributions of satellites sur-rounding the cores of the haloes to confirm the existence of a pref-erential plane for the satellite locations in projection. In Fig. 11, we show the compilation of the projected positions of satellites in the core’s rest frame. The result is a synthetic galactic system with 100 000 satellites in the same rest frame. We performed suitable ro-tations to bring the spin axis collinear to the z-axis for each system of satellites, and then we added all these systems to obtain the actual synthetic halo with 100 000 satellites. The positions were normal-ized using the mother’s radius (which is of the order of the virial radius). A quick analysis of the isocontours of the satellite

distri-−1 0 1 −1 0 1 y/Rm z/R m −1 0 1 −1 0 1 y/Rm z/R m

Figure 11. The projected distribution of satellites around the core’s centre

of mass. We used the position of 40 000 satellites around their respective core to produce a synthetic halo plus satellites (a ‘mother’) system. The projection is performed along the x-axis. The y and z coordinates are given in units of the mother’s radius. The z-axis is collinear to the direction of the core’s spin. Top: the isocontours of the number density of satellites around the core’s centre of mass present a flattened shape. The number of satellites is lower in darker bins than in lighter bins. The flattened isocontours indicate that satellites lie preferentially in the plane orthogonal to the direction of the spin. Bottom: the excess number of satellites surrounding the core. We compared the distribution of satellites measured in our simulations to an isotropic distribution of satellites. Light zones stand for an excess of satellites in these regions (compared to an isotropic distribution) while dark zones stand for a lack of satellites. The satellites are more numerous in the equatorial region than expected in an isotropic distribution of satellites around the core. Also, there are fewer satellites along the spin’s axis than expected for an isotropic distribution of satellites.

butions indicates that satellites are more likely to be found in the equatorial plane, even in projection. The axial ratio measured at one mother’s radius is(Rm)≡ a/b − 1 = 0.1 with a > b. We

compared this distribution to an isotropic ‘reference’ distribution of satellites surrounding the core. This reference distribution has the same average radial profile as the measured satellite distributions but with isotropically distributed directions. The result of the sub-traction of the two profiles is also shown in Fig. 11. The equatorial plane (perpendicular to the z-axis) presents an excess in the number of satellites (light regions). Meanwhile, there is a lack of satellites along the spin direction (dark regions). This confirms our earlier results obtained using the alignment of orbital momentum of satel-lites with the core’s spin, i.e. satelsatel-lites lie more likely in the plane

orthogonal to the halo’s spin direction. Qualitatively, these results have already been obtained by Tormen (1997), where the major axis of the ellipsoid defined by the satellites’ distribution is found to be aligned with the cluster’s major axis. This synthetic halo is more di-rectly comparable to observables since, unlike the dark matter halo itself, the satellites should emit light. Even thoughCDM predicts too many satellites, its relative geometrical distribution might still be correct. In the following sections, our intent is to quantify this effect more precisely.

The propensity of satellites to lie in the plane orthogonal to the direction of the core’s spin appears as an ‘anti-Holmberg’ effect. Holmberg (1974) and more recently Zaritsky et al. (1997) have found observationally that the distribution of satellites around discs is biased towards the pole regions. Thus if the orbital momentum vector of galaxies is aligned with the spin of their parent haloes, our result seems to contradict these observations. One may argue that satellites are easier to detect out of the galactic plane. Furthermore our measurements are carried far from the disc while its influence is not taken in account. Huang & Carlberg (1997) have shown that the orbital decay and the disruption of satellites are more efficient for coplanar orbits near the disc. This would explain the lack of satellites in the disc plane. Thus our distribution of satellites can still be made consistent with the ‘Holmberg effect’.

5.2 Projected satellite orientation and spin

In addition to the known alignment on large scales, we have shown that the orientation of structures on smaller scales should be different from that expected for a random distribution of orientations. Can this phenomenon be observed? The previous measurements were carried in 3D while this latter type of observation is performed in projection on the sky. The projection ‘dilutes’ the anisotropy effects detected using 3D information. Thus an effect of 15 per cent may be lowered to a few per cent by projecting on the sky. However, even if the deviation from isotropy is as important as a few per cent, as we will suggest, this should be relevant for measurements involved in extracting a signal just above the noise level, such as weak lensing. To see the effect of projection on our previous measurements, we proceed in two steps. First, every mother (halo core plus satel-lites) is rotated to bring the direction of the core’s spin to the z-axis. Secondly, every quantity is computed using only the y and z com-ponents of the relevant vectors, corresponding to a projection along the x-axis.

The first projected measurement involves the orientation of satel-lites relative to their position in the core’s rest frame. The spin of a halo is statistically orthogonal to the main axis of the distribution of matter of that halo (Faltenbacher et al. 2002), and assuming that this property is preserved for satellites, their spin Ssis an

indica-tor of their orientation. The angle,θP(in projection), between the

satellites’ spin and their position vector (in the core’s rest frame) is computed as follows: θP= cos−1  S_sy,z·ry,z sc S_sy,zrscy,z  , (30) with rsc= rs− rc, (31)

where rs and rcstand respectively for the position vector of the

satellite and the core’s centre of mass. Two extreme situations can be imagined. The ‘radial’ configuration corresponds to a case where the satellite’s main axis is aligned with the radius joining the core’s centre of mass to the satellite centre of mass (spin perpendicular

proj. angles distrib.

gaussian fit 0.96 0.98 1.00 1.02 1.04 Θ (rad) 1+ ξ (Θ )

0

π/4

π/2

3

π/4

π

Figure 12. Excess probability, 1+ ξP, of the projected angles between the

direction of the spin of substructures and their position vector in the core’s rest frame. The projection is made along the x-axis where the z-axis is co-incident with the core’s spin direction. The solid line represents the average distribution of projected angles of 50 subsamples of 50 000 substructures (out of 100 000 available substructures). The error bars represents the 3σ dispersion relative to these 50 subsamples. An isotropic distribution of ori-entation would correspond to a value of 1 for 1+ ξP. The projection plus

the reference to the position vector instead of the velocity’s direction lowers the anisotropy effect. The dashed curve stands for the best Gaussian fit of the excess probability (see equation 13 for parametrization). The best-fitting pa-rameters are: a1= 0.0999 ± 0.0030, a2= 1.5488 ± 0.0031, a3= 0.8259 ±

0.0131 and a4= 0.9737 ± 0.0007. It seems that on average the projected

orientation of a substructure is orthogonal to its projected position vector.

to the radius, orθP∼ π/2). The ‘circular’ configuration is the case

where the satellite main axis is orthogonal to the radius (spin parallel to the radius,θP∼ 0 [π]). These reference configurations will be

discussed in what follows.

The resulting distribution, 1 + ξP(θP), is shown in Fig. 12.

As before, an isotropic distribution of orientations would lead to

ξP(θP)= 0. The distribution is computed with 100 000 satellites,

without the cores, while the error bars result from Monte Carlo sim-ulations on 50 subsamples of 50 000 satellites each. As compared to the distribution expected for random orientations, the orthogonal configuration is present in excess ofξP(π/2) ∼ 0.02. If the spin of

satellites is orthogonal to their principal axis, the direction vector in the core’s rest frame is more aligned with the satellites’ principal axes than one would expect for an isotropic distribution of satellite orientations. This configuration is ‘radial’. The peak of the distri-bution is slightly above the error bars:ξP(θP∼ π/2) ∼ 0.02. The

distribution can be fitted by the Gaussian function given in equa-tion (13) with the following parameters: a1 = 0.0999 ± 0.0030,

a2= 1.5488 ± 0.0031, a3= 0.8259 ± 0.0131 and a4= 0.9737 ±

0.0007. The alignment seems to be difficult to detect in projection. With 50 000 satellites, we barely detect the enhancement of the or-thogonal configuration at the 3σ level, and thus we do not expect a detection of this effect at the 1σ level for less than 6000 satellites. Nevertheless, the distribution of the satellites’ orientation in projec-tion seems to be ‘radial’ on dynamical grounds, without reference to a lensing potential.

Our previous measurement was ‘global’ since it does not take into account the possible change of orientation with the relative position of the satellites in the core’s rest frame. In Fig. 13, we